This study will test the hypothesis that reconstitution of immune function in adenosine deaminase (ADA)-deficient severe combined immunodeficiency syndrome (SCID) will result from the autologous transplantation of CD34+ umbilical cord blood stem cells that have been transduced with a normal human ADA complementary DNA by retroviral-mediated gene transfer. The specific aims of this protocol are to: 1)determine whether transduced CD34+ umbilical cord blood stem cells will engraft following autologous transplantation in an infant with ADA deficiency; 2) determine the extent and durability of engraftment in different hematopoietic cell populations; 3) determine the extent and durability of immune reconstitution following gene therapy.